Multiple Bifurcations in a Polynomial Model of Bursting Oscillations

Bursting oscillations are commonly seen to be the primary mode of electrical behaviour in a variety of nerve and endocrine cells, and have also been observed in some biochemical and chemical systems. There are many models of bursting. This paper addresses the issue of being able to predict the type of bursting oscillation that can be produced by a model. A simplified model capable of exhibiting a wide variety of bursting oscillations is examined. By considering the codimension-2 bifurcations associated with Hopf, homoclinic, and saddle-node of periodics bifurcations, a bifurcation map in two-dimensional parameter space is created. Each region on the map is characterized by a qualitatively distinct bifurcation diagram and, hence, represents one type of bursting oscillation. The map elucidates the relationship between the various types of bursting oscillations. In addition, the map provides a different and broader view of the current classification scheme of bursting oscillations.

[1]  M. Steriade,et al.  Thalamic bursting mechanism: an inward slow current revealed by membrane hyperpolarization , 1982, Brain Research.

[2]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[3]  J. L. Hudson,et al.  An experimental study of multiple peak periodic and nonperiodic oscillations in the Belousov–Zhabotinskii reaction , 1979 .

[4]  John Rinzel,et al.  Bursting oscillations in an excitable membrane model , 1985 .

[5]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[6]  Teresa Ree Chay,et al.  Endogenous Bursting Patterns in Excitable Cells , 1988 .

[7]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[8]  P M Dean,et al.  Glucose‐induced electrical activity in pancreatic islet cells , 1970, The Journal of physiology.

[9]  E. Kandel,et al.  Electrophysiology of hippocampal neurons. II. After-potentials and repetitive firing. , 1961, Journal of neurophysiology.

[10]  R. Bertram,et al.  Topological and phenomenological classification of bursting oscillations. , 1995, Bulletin of mathematical biology.

[11]  J. Sotomayor,et al.  GENERIC 3-PARAMETER FAMILIES OF PLANAR VECTOR-FIELDS, UNFOLDINGS OF SADDLE, FOCUS AND ELLIPTIC-SINGULARITIES WITH NILPOTENT LINEAR PARTS , 1991 .

[12]  P. Schwartzkroin,et al.  Electrophysiology of Hippocampal Neurons , 1987 .

[13]  John Rinzel,et al.  A Formal Classification of Bursting Mechanisms in Excitable Systems , 1987 .

[14]  Mark Pernarowski,et al.  Fast Subsystem Bifurcations in a Slowly Varying Liénard System Exhibiting Bursting , 1994, SIAM J. Appl. Math..

[15]  J. Guckenheimer,et al.  Bifurcation of the Hodgkin and Huxley equations: A new twist , 1993 .

[16]  R K Wong,et al.  Afterpotential generation in hippocampal pyramidal cells. , 1981, Journal of neurophysiology.

[17]  John Guckenheimer,et al.  Multiple bifurcation problems for chemical reactors , 1986 .

[18]  John Guckenheimer,et al.  Multiple Bifurcation Problems of Codimension Two , 1984 .

[19]  M. Kim,et al.  Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. , 1976, Biophysical journal.

[20]  F. Ashcroft,et al.  Electrophysiology of the pancreatic beta-cell. , 1989, Progress in biophysics and molecular biology.

[21]  A. Goldbeter,et al.  From simple to complex oscillatory behaviour: analysis of bursting in a multiply regulated biochemical system. , 1987, Journal of theoretical biology.

[22]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[23]  T. Erneux,et al.  Understanding bursting oscillations as periodic slow passages through bifurcation and limit points , 1993 .

[24]  G. Ermentrout,et al.  Parabolic bursting in an excitable system coupled with a slow oscillation , 1986 .

[25]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.